Hadamard matrices are used for improving precision in measurements.

For instance, the columns may be interpreted as weighings on an instrument with two scales, the rows as objects to be weighed. During each of the n weighings, all n objects are being used. In weighing j object i is put on the left scale if the entry H_ij = 1 and on the right otherwise.

This way of weighing will give more accurate (less variance) weight measurements than when they are being weighed individually.

Let H be an n×n Hadamard matrix. The rows of H and -H are the code words of an error correcting code.

Suppose c is a code word, then cH^T is a vector with all entries 0 except for one, which is equal to n or -n. This exception occurs at the coordinate which is the row number of c or -c in H. Suppose this row number is m.

Suppose now that we make a mistake and do not find the word c but make an error by switching some of the signs of the entries in c. Then we find a word of the form c + e, where e is the vector consisting of 0 at the places where we did not make an error and of -2 or +2 at the places where we did make an error.

Compute (c + e)H^T. This product (c + e)H^T differs eH^T from cH^T. If the number of errors equals k, and k < n/4, then the m-th coordinate of (c + e)H^T has absolute value at least n - 2k > n/2, while all other coordinates have absolute value at most 2k < n/2. In particular, we can recover m from this product and hence, as there are at most k < n/2 errors in c + e, we can recover the vector c as the m-th row of H or -H.

Concluding, we notice that the code consists of 2n code words of length n and that we can correct at least n/4 - 1 errors.

A Hadamard matrix of dimension 32 was actually used by the Mariner, which traveled to Mars in 1969, to encode pictures of that planet and send them to the earth.